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f4b3ad1 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.NumberTheory.Ostmann.Arithmetic.BulkSlotNodup
import OAI.NumberTheory.Ostmann.Arithmetic.FrozenFrequencyAverage
namespace OAI
/-! # The original arithmetic support forces distinct bulk primes -/
namespace Ostmann
open scoped Classical BigOperators
theorem prime_list_injective_of_pairwise {σ : Type*} (value : σ → ℕ)
(L : List σ) (hprime : ∀ i ∈ L, (value i).Prime)
(hp : (L.map value).Pairwise Nat.Coprime) :
∀ i ∈ L, ∀ j ∈ L, value i = value j → i = j := by
have hn : (L.map value).Nodup := by
apply hp.imp_of_mem
intro a b ha hb hab
obtain ⟨i, hi, rfl⟩ := List.mem_map.mp ha
obtain ⟨j, hj, rfl⟩ := List.mem_map.mp hb
exact (Nat.coprime_primes (hprime i hi) (hprime j hj)).mp hab
exact fun i hi j hj hij => List.inj_on_of_nodup_map hn hi hj hij
theorem buildMovingSlotData_injective_of_support {σ : Type*}
(value : σ → ℕ) (n m : ℕ) (slot : (TreeLeafIndex n × Fin m) ↪ σ)
(t : FrequencyTree ℤ n) (small : TreeLeafTuple (List σ) n)
(samples : MovingSampleSlots σ n) (outside : List ℕ)
(hprime : ∀ i, (value i).Prime)
(hs : movingRegularOutsidePairwise value outside
(buildMovingSlotData n t small (bulkSlotLeaves n m slot) samples)) :
Function.Injective (value ∘ slot) := by
let T := buildMovingSlotData n t small (bulkSlotLeaves n m slot) samples
have hp : (T.regularSlots.map value).Pairwise Nat.Coprime := by
have hroot : (T.regularSlots.map value ++ outside).Pairwise Nat.Coprime := by
cases hT : T with
| leaf s L => simpa only [T, hT, movingRegularOutsidePairwise,
MovingSlotData.regularSlots] using hs
| node s CL CR u left right =>
have hs' := hs
change movingRegularOutsidePairwise value outside T at hs'
rw [hT] at hs'
exact hs'.1
exact (List.pairwise_append.mp hroot).1
have hm (j) : slot j ∈ T.regularSlots := by
apply (buildMovingSlotData_regular_perm n t small (bulkSlotLeaves n m slot) samples).mem_iff.mpr
exact List.mem_append_right _ ((mem_flatten_bulkSlotLeaves n m slot _).mpr ⟨j, rfl⟩)
intro i j hij
exact slot.injective (prime_list_injective_of_pairwise value T.regularSlots
(fun i _ => hprime i) hp (slot i) (hm i) (slot j) (hm j) hij)
/-- The frequency factor vanishes on collisions before any averaging or
resampling is performed. -/
theorem movingFrequencyPageAverage_zero_of_bulk_collision {σ : Type*}
(value : σ → ℕ) (n m : ℕ) (slot : (TreeLeafIndex n × Fin m) ↪ σ)
(t : FrequencyTree ℤ n) (small : TreeLeafTuple (List σ) n)
(samples : MovingSampleSlots σ n) (outside : List ℕ)
(hprime : ∀ i, (value i).Prime) (hbad : ¬Function.Injective (value ∘ slot))
(F : Bool → {k : ℕ} → MovingSlotData σ k → ℤ → ℂ)
(E : Bool → {k : ℕ} → MovingSlotData σ k → ℤ → ℤ → ℤ → ℝ)
(T : Bool → MovingSlotData σ n)
(hT : T false = buildMovingSlotData n t small (bulkSlotLeaves n m slot) samples)
(nodes : Bool → List MovingFormulaNode) (R : ℤ)
(r : ℕ) [NeZero r] (input : PublishedProgressionInput) (Q : ℕ) (y : ℝ) :
movingFrequencyPageAverage value outside F E T nodes R r input Q y = 0 := by
have hsupport : ¬(∀ b, movingRegularOutsidePairwise value outside (T b) ∧
∀ f ∈ nodes b, f.guard.frequencyBounds) := by
intro hs
apply hbad
apply buildMovingSlotData_injective_of_support value n m slot t small samples outside hprime
rw [← hT]
exact (hs false).1
rw [movingFrequencyPageAverage_core]
simp only [hsupport, ite_false, zero_mul]
/-- Inserting the distinctness gate leaves the literal supported arithmetic
coefficient unchanged. -/
theorem movingFrequencyPageAverage_collision_gate {σ : Type*}
(value : σ → ℕ) (n m : ℕ) (slot : (TreeLeafIndex n × Fin m) ↪ σ)
(t : FrequencyTree ℤ n) (small : TreeLeafTuple (List σ) n)
(samples : MovingSampleSlots σ n) (outside : List ℕ)
(hprime : ∀ i, (value i).Prime)
(F : Bool → {k : ℕ} → MovingSlotData σ k → ℤ → ℂ)
(E : Bool → {k : ℕ} → MovingSlotData σ k → ℤ → ℤ → ℤ → ℝ)
(T : Bool → MovingSlotData σ n)
(hT : T false = buildMovingSlotData n t small (bulkSlotLeaves n m slot) samples)
(nodes : Bool → List MovingFormulaNode) (R : ℤ)
(r : ℕ) [NeZero r] (input : PublishedProgressionInput) (Q : ℕ) (y : ℝ) (A : ℂ) :
movingFrequencyPageAverage value outside F E T nodes R r input Q y * A =
if Function.Injective (value ∘ slot) then
movingFrequencyPageAverage value outside F E T nodes R r input Q y * A else 0 := by
by_cases h : Function.Injective (value ∘ slot)
· simp only [h, ite_true]
· rw [ite_eq_right h, movingFrequencyPageAverage_zero_of_bulk_collision value n m slot
t small samples outside hprime h F E T hT nodes R r input Q y, zero_mul]
end Ostmann
end OAI
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